Regarding conditions required and specific topology to be induced on borel sets to define a specific measure

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I am trying to solve assignment problems of an institute in which I don't study as our faculty doesn't gives assignments and I'm unable to think about how to proceed through this particular problem.

Problem is ----- Define extra conditions on Borel sets and also tell the underlying topology on borel sets, so as to extract a measure such that $ \mu $ (E) = 0 for every Borel Set E in the topology induced?

I have studied First two chapter of Walter Rudin real and complex analysis and GB Folland a guide to advanced real analysis but I could not think about how to solve this.

Please give some hints.